# Let x1 x2 be iid with exi let sn s0 x1

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Unformatted text preview: ibuted amount of time, ﬁnd the stationary distribution of the number of children playing or in line at each of the two machines. 158 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Chapter 5 Martingales In this chapter we will introduce a class of process that can be thought of as the fortune of a gambler betting on a fair game. These results will be important when we consider applications to ﬁnance in the next chapter. In addition, they will allow us to give more transparent proofs of some facts from Chapter 1 concerning exit distributions and exit times for Markov chains. 5.1 Conditional Expectation Our study of martingales will rely heavily on the notion of conditional expectation and involve some formulas that may not be familiar, so we will review them here. We begin with several deﬁnitions. Given an event A we deﬁne its indicator function ( 1 x2A 1A = 0 x 2 Ac In words, 1A is “1 on A” (and 0 otherwise). Given a random variable Y , we deﬁne the integral of Y over A to be E (Y ; A) = E (Y 1A ) Note that multiply...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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